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Math Help - Find the sum of the convergent series?

  1. #1
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    Find the sum of the convergent series?

    I figured an equation that works for this series but have no idea how to calculate it once I have the sum...none of the rules in our book seem to apply.

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  2. #2
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    Quote Originally Posted by CrazyLond View Post
    I figured an equation that works for this series but have no idea how to calculate it once I have the sum...none of the rules in our book seem to apply.

    Try to use some algebra

     \left( \frac{1}{2}\right)^{2n-5}=\left( \frac{1}{2}\right)^{2(n-1)-3}=\left( \frac{1}{4}\right)^{n-1} \left( \frac{1}{2}\right)^{-3} =8\left( \frac{1}{4}\right)^{n-1}

    So your sum is the same as

     8\sum_{n=1}^{\infty}\left( \frac{3}{4} \right)^{n-1}

    So r=\frac{3}{4}

    So the sum of the series 8\cdot\frac{1}{1-\frac{3}{4}}=8\cdot\frac{1}{\frac{1}{4}}=32

    P.S. I think your index starts at n=1 not at n =0.
    Last edited by TheEmptySet; June 1st 2008 at 08:47 PM. Reason: wrote the problem down wrong
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  3. #3
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    Hello, CrazyLond!

    S \;=\;8 + 6 + \frac{9}{2} + \frac{27}{8} + \hdots

    Geometric series: first term a = 8, common ratio r = \frac{3}{4}

    Its sum is: . S \;=\;\frac{8}{1 - \frac{3}{4}} \;=\;32

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  4. #4
    Super Member angel.white's Avatar
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    Quote Originally Posted by CrazyLond View Post
    I figured an equation that works for this series but have no idea how to calculate it once I have the sum...none of the rules in our book seem to apply.

    I find it easier to look at these like this:

    a_0 = 8

    a_n = \frac 34a_{n-1}

    Looking at that a little bit (you can do the first few a_n if it helps to see, then you will be familiar enough to spot it right off) it should be clear that a_n = 8*\left(\frac 34\right)^n

    And since you want to sum all of them, you get

    \sum_{n=0}^\infty a_n

    =\sum_{n=0}^\infty 8*\left(\frac 34\right)^n

    =8\sum_{n=0}^\infty \left(\frac 34\right)^n

    At which point you have the same geometric series that TheEmptySet and Soroban solved for.
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